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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

14158 questions
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Newton Method convergence

Want to apply Newtons Method to solve f(x)=0 where f is such that |f ''(x)|≤ 10 and |f ' (x)| ≥ 2 for all x. How do I explain that $x_0$ can take any positive starting value for the method to converge i.e $x_0$ does not need to be specifically close…
Keralf2020
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Why does the backward recursion work better?

Computing the following integral: \begin{equation*} I_n = \int_0^1 x^n e^x \: dx, \end{equation*} using the forward recursive formula \begin{align} I_0 &= e-1 \\ I_{n} &= e - n I_{n-1} \end{align} is an unstable calculation. Here we will consider a…
Cohen
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Numerical Solution to $ 7000 = \sum_{n=1}^{5} \frac{ \ 1000}{(1 + x)^n} + \sum_{n=6}^{10} \frac{\ 2000}{(1 + x)^n}. $

$$ 7000 = \sum_{n=1}^{5} \frac{ \ 1000}{(1 + x)^n} + \sum_{n=6}^{10} \frac{\ 2000}{(1 + x)^n}. $$ I need to solve the equation above. Since it generates a $10th$ degree polynomial, the equation must be solved by some numerical method. However, I'm…
Sigma
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Prove that $f[x_0,x_0,x_1]=f'[x_0,x_1]$ for divided difference

Prove that $f[x_0,x_0,x_1]=f'[x_0,x_1]$ for divided difference Attempt RHS \begin{align*} f'[x_0,x_1]&=\lim_{\epsilon\rightarrow 0}\frac{f[x_0+\epsilon,z_1]-f[x_0,x_1]}{\epsilon} \end{align*} LHS…
user1942348
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Converging cubically

Consider $F(x)=x-f(x)f'(x)$ where for some $r$ $f(r)=0, 0\neq f'(r)$. Find a perciese condition of $f$ such that $x_{k+1}=F(x_k)$ converge to the fixed-point $r$ at least cubically if started near $r$. I want somehow to show $|g(x)-g(y)|
Ahmed
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Improving the convergence of the series

I am trying to calculate the sum of a A = $\sum_{1}^{\infty}\frac{1}{n^2 + 0.4}$ with precision $0.5\cdot 10^{-7}$ $R_{method} = |A - S_{N}| = \sum_{N+1}^{\infty}\frac{1}{n^2 + 0.4}$ $\sum_{N+1}^{\infty}\frac{1}{n^2 + 0.4} <…
Sh VavilenT
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Evaluating solutions numerically to the following equations

How would one use numerical methods to find solutions to the following…
hwood87
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How to deal with $f(z_{k+1})$ term?

The method for roots of a scalar function $f$ alternates Newton steps ($x_k\mapsto z_{k+1}$) and simplified Newton steps ($z_{k+1}\mapsto x_{k+1}$). I think I need to Taylor expand all the f, but I do not know how to deal with the $f(z_{k+1})$…
Snuggles
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Can I repeatedly solve $Ax=b$ where $A$ is a function $x$ by using the matrix $A$ evaluated at the previous value of $x$ to solve for $x?$

I want to know if this method is sure to converge. I have a feeling that this is already a well-documented method. If that is the case, can somebody please mention the name of this method? Any help is much appreciated. PS: I am trying to solve a…
Utkarsh Singh
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Searching for a symbolic expression

I know that there was a "calculator" that when given a decimal number, it would output the most likely candidate for an exact expression for this number. So, for example, if one were to input $1.414$, the "calculator" may say $\sqrt 2$. However, I…
FusRoDah
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Numerical integration: example

Consider $I = \int_a^b f(x) dx = 267.25$ and the approximations $\overline{I}_8 = 295.27$, $\overline{I}_{16} = 274.15$, $\overline{I}_{32} = 268.97$ and $\overline{I}_{64} = 267.68$ of $I$ given by the composite Simpson's rule with $9,17,33$ and…
Nick
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Calculating real roots and multiplicity

How can I calculate real roots as accurate as possible and find the multiplicity of each root for $$p(x)=10.0x^5+9.0x^4-16.2x^3-14.58x^2+6.561x+5.9049?$$ I know there are methods like bisection, secant, inverse-quadratic interpolation or Brent's…
user60514
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Numerical Integration over 3D mesh.

I have a 3D mesh composed of tessellating triangles. I would like to perform an integral over the surface of this mesh of the form: $$\int f(\Omega)R(\Omega)d\Omega$$ Where $R(\Omega)$ is the radius at a given angle and $f(\Omega)$ is an arbitary…
user668074
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Calculate variable in expression with Truncate function

I would like to know how to calculate MaxDigitWidth variable by using CharWidth and Pixels variables in the following expression: CharWidth = Truncate((Pixels - 5) / MaxDigitWidth * 100 + 0.5) / 100 It should be in a form: MaxDigitWidth = ...…
Stipo
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How are weights determined in a numerical scheme?

A numerical scheme has linear\convex combinations to weight the slopes at different time steps. How are these weights determined? It is through some mathematical techniques or through trial & error? This example has the description and algorithm for…
spicy
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